A chargeback is a demand by a credit-card provider to make good the loss on a fraudulent or disputed transaction. It is between the merchant and the card-issuing bank (usually on behalf of the cardholder). The arbitrator is a representative from the card brand association. It is strictly a document review process so there are no hearings to be present at either in person or by phone.
Each association is a little different with their terminology for each stage but the general process flow is pretty consistent. The cardholder reaches out to the card-issuing bank to dispute a payment. The merchant receives written notification of said dispute and has the option to accept the dispute or submit a response explaining why it was a valid charge. If a response is submitted and the cardholder wins, it ends there. If the merchant wins, however, the card-issuing bank can challenge the decision. This is what happens when you see the same chargeback come through a second time. Everything stays in the file so anything new you send in when responding to a second chargeback just adds to what is already there. This means that your second response needs to add stronger, more compelling evidence. It does not work to simply resend the paperwork submitted for the first response.
Once you are presented with the option to argue a second chargeback, it is important to consider the chargeback reason. Any time the reason is fraud and the issuer maintains the charge was not authorized, it is very hard for the merchant to win. You also want to carefully weigh the amount of the transaction being disputed to see if it is worth the potential total loss in fees. As a case escalates, you could incur fees up to around $500 if you keep submitting responses and ultimately do not win the case. This is in addition to the nominal occurrence fees your merchant account provider charge for each retrieval and chargeback.
Finally, each transaction has it's own case number and each case is reviewed individually. If you have a client disputing multiple payments, this process can become very cumbersome to manage. When you submit a response, you should not assume that details from one case will carry over to another just because it is the same cardholder so you will need to repeat yourself. You can and should reference other cases for the same client if it will help your case.